@famesyasd: Well I'd have to say your statements are ambiguous, which is why @AlessandroCodenotti is not very satisfied haha..
I would state it as "f is continuous at c iff ( given any parameter x in the domain of f we have f(x) ≈ f(c) as x → c )."
One could define "f has limit L near c iff ( given any parameter x in the domain of f we have f(x) ≈ L as x → c )."
Both of these could be interpreted via the standard δ-ε formulation, or via sequences, or simply in an axiomatic framework.
For example, here's the sequence formulation:
"f is continuous at c iff ( given any sequence x from the domain of f we have f(x[n]) ≈ f(c) as natural n → ∞ )."
If it looks too similar, it's deliberate. Of course some people write things like "lim[n→∞] f(x[n]) = f(c)", but that's actually technically ambiguous because it doesn't specify what kind of entity n is.
But anyway, yes the point is that f is continuous at c iff f has limit f(c) at c.
Anyway I'm going off soon. See you both next time! =)